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On Level Clustering in Regular Spectra 数理解析研究所講究録 1156 巻 2000 年 113-130 113 On Level Clustering in Regular Spectra Nariyuki MINAMI* $l’\phi'/|\tau\dot{L}^{\prime \mathrm{f}^{\nearrow}\prime}'\dot{\eta}'$ ) lnstitute of Mathematics, University of Tsukuba Tsukuba-shi, lbaraki, 305-8571 Japan 1 Introduction. Let $H(\hslash)$ be the quantum Hamiltonian describing a bounded, isolated physical system of finite degrees of freedom. ( $\hslash$ is the Planck’s constant.) Then the spectrum of $H(\hslash)$ is purely discrete, consisting of energy levels $\{E_{n}(\hslash)\}_{n\geq 1}$ . Suppose further that $H(\hslash)$ , or its spectrum itself, is obtained by quantizing (in a certain way) the classical Hamiltonian $H(p, q)$ defining the system. Now the classical dynamical system corresponding to $H(p, q)$ may belong to one of two extreme cases of being completely integrable or chaotic. It was Percival [12] who proposed to distinguish the spectrum of $H(\hslash)$ according to the category to which the classical counterpart of $H(\hslash)$ belongs. He called the spectrum regular [resp. irregular] if $H(p, q)$ is integrable [resp. chaotic], and discussed possible difference between these two kinds of spectra. Later, Berry and Tabor [2] argued that regular and irregular spectra are distinguished by looking at the probability distribution of the spacing between adjacent energy levels. Namely they argued that in the regular spectra, the level spacing obeys the exponential distribution $e^{-t}dt$ , so that the spectrum looks like a typical realization of the Poisson point process, which is the phenomenon they called “level clustering ”. On the other hand, they conjectured that in the irregular spectra, the level spacing distribution $p(t)dt$ satisfies $p(t)\sim \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{S}\mathrm{t}.t^{\gamma},$ $t\searrow \mathrm{O}$ , with $\gamma>0$ (“level repulsion ”). Hence irregular spectra should typically look like the spectra of large random matrices. This conjecture is supported by numerical studies performed later. (See e.g. [3] for a review.) However, neither the precise meaning of the “probability distribution ”of level spacing, nor a mathematical formulation of the statistics for the spectrum of $H(h)$ -the level statistics-in general seems to have been explicitly given in physics litterature, although $\mathrm{s}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{i}\mathrm{l}\mathrm{n}\mathrm{p}_{0}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$ ideas are stated in [2]. The present paper, which is an elaboration of a part of the author’s previous note [10], aims at giving a mathematical formulation of level statistics based on the idea of Berry and Tabor, and applying it to regular spectra. In \S 2, we give the definition of strict (Definition 1) and wide (Definition 2) sense level clustering, and prove solne preliminary results for later references. These results are formulated in analogy with corresponding propositions in the theory of stationary point processes. In \S 3, we shall apply the level statistics thus formulated to regular spectra, and discuss $\mathrm{t}1_{1}\mathrm{e}$ closely related theorems by Sinai [15] and Major [7]. We argue that although it is probably very difficult to apply theorems of Sinai and Major to prove the strict sense $*$ -mail. [email protected] 114 level clustering in gelleric regular spectra, there is solne hope in proving the wide sense level clustering for solne concrete Hamiltonian such as rectangular billiard. Finally in \S 4, we shall propose another fornmlation of level statistics, and with an exalllple, shall argue again that “level clustering ”should not always mean strict Poissonian property of the $\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{C}\mathrm{t}_{\Gamma}\mathrm{u}\mathrm{l}\mathrm{n}$ . 2 Strict and wide sense level clustering. 2.1 The unfolding of Berry and Tabor. When one speaks of the “probability distribution ”of the spacing of the adjacent ellergy levels of $H(\hslash)$ , the follwing two questions immediately arise: 1. There is no stochasticity in $H(\hslash)$ , hence in $\{E_{l},(h)\}_{n\geq 1}$ too. $\mathrm{T}\mathrm{h}\mathrm{e}1^{\cdot}\mathrm{f}_{0}\mathrm{r}\mathrm{e}$ . it will $\mathrm{b}`\lrcorner$ $\mathrm{t}1_{1}\mathrm{e}$ necessary to take semiclassical limit $h\searrow \mathrm{O}$ to have sufficiently many levels in a fixed energy intervals, and we will have to take statistics among these levels. $\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\ln$ 2. The lnean level density is not over the entire $\mathrm{s}_{\mathrm{P}^{\mathrm{e}\mathrm{C}\mathrm{t}\mathrm{r}\mathrm{U}}}111$ , so that the statistical property of $E_{n+1}-E_{n}$ may not be uniform in $n$ . Hence we will need to $\mathrm{u}\mathrm{n}\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{c}|$ ”the spectrunl so that it will look like a $\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}_{\mathrm{o}\mathrm{r}}1\mathrm{I}\mathrm{l}\mathrm{l}\mathrm{y}$ distributed sequence. $\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{n}\mathrm{l}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}$ lnspired by [2], we make a of the $\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}_{\Gamma}\mathrm{u}111\{E_{l},(h)\}_{\iota\geq},1$ of $H(\hslash)\backslash \mathrm{v}\mathrm{h}\mathrm{i}(.11$ lneets the above two $\mathrm{r}\mathrm{e}\mathrm{q}_{\mathrm{U}\mathrm{i}\mathrm{e}\mathrm{l}\mathrm{I}1}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}$ simultaneously. For this purpose, we lnake the following $\mathrm{a}\mathrm{S}\mathrm{s}\mathrm{u}111\mathrm{P}^{\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}}$ : (A1) All levels are non-degenerate and $E_{n}(\hslash)\geq 0$ ; (A2) $E_{n}(h)\searrow 0$ monotonically as $h\searrow 0$ ; (A3) For each fixed $E>0$ , there is a constant $l\text{ノ}(E\mathrm{I}>0$ such that $N_{\hslash}(E)\equiv\#\{n\geq 1|E_{n}(\hslash)\leq E\}\sim\iota \text{ノ}(E)h^{-}d(h\searrow 0)$ . (1) Let us take $E>0$ , which we shall fix throughout. By (A1) and (A2), we can define $\Gamma_{l},\cdot=h_{j}(E)$ as the unique solution of the equation $E_{j}(\hslash)=E$ . If we define $\lambda_{j}=\iota \text{ノ}(E\mathrm{I}\hslash,\cdot(E)-d$ , (2) $\mathrm{t}1_{1\mathrm{e}\mathrm{n}}\mathrm{f}_{\Gamma}\mathrm{o}111$ (A3), it is easily seen that $n(L)\equiv\#\{j\geq 1|\lambda_{\dot{j}}\leq L\}\sim L$ $(Larrow\infty)$ . (.3) Thus we have obtained a sequence $\{\lambda\}_{j}$ which has asymptotically uniforlIl distribution. We will call the $\lambda_{j}' \mathrm{s}$ the unfolded levels, and call the above procedure $\zeta(\mathrm{t}\mathrm{h}\mathrm{e}$ unfolding of $‘$ ( Berry and Tabor ”. (It is also called $\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}$ of Planck’s constant ”in [2].) 115 2.2 Statistics for asymptotically uniformly distributed sequences. $\mathrm{t}1_{1}\mathrm{i}\mathrm{s}$ In subsection, we suppose that $\{\lambda,\}_{=0}^{\infty},\cdot$ is simply a strictly increasing sequence of real ( 11nlIlbers satisfying (3) and $\lambda_{0}=0$ , but still call $\lambda_{j}$ the $‘ j$ -th level ”. We introduce the $\mathrm{f}\mathrm{o}1]0\backslash \mathfrak{j}\prime \mathrm{i}_{1\mathrm{l}}\mathrm{g}$ notation: . $\mathit{1}j\backslash ^{\tau}(t)=N(t,\cdot c)=\#\{j\geq 1|\lambda_{j}\in(t, t+c]\}=,\cdot\sum_{\geq 1}1_{(t.\mathrm{f}}+c](\lambda i))$ (4) $\pi_{k}(c;L)=\frac{1}{L}\int_{0}^{L}.1_{\{()}Nt:c=k\}dt$ ; (5) $\mu_{k}(c;L)=\frac{1}{L}\int_{0}^{L}dt=,\cdot\sum_{\geq k}\pi_{j}(_{C},\cdot L)$ , (6) $\backslash \iota' 1\mathrm{l}\mathrm{e}\mathrm{l}\cdot \mathrm{e}$ $= \frac{1}{k!}j(j-1)\cdots(j-k+1)$ ; (7) ’ dlld $\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{l}\backslash ./$ . $\rho(c;L)=\frac{\#\{_{J}\geq 1|\lambda,\cdot\leq L,\lambda,+1-\lambda_{j}\leq C\}}{\#\{j\geq 1|\lambda_{j}\leq L\}}$ (8) Here $c>0$ and $k=0,1,2,$ $\ldots$ . These notions have the following obvious probabilistic meanings. $N(t;c)$ is nothing but the number of levels in the interval $(t, t+c]$ . $\pi_{k}.(c;L)$ is the probability that this $\mathrm{i}_{11\mathrm{t}1}\mathrm{e}\cdot\backslash \cdot \mathrm{a}1$ catches exactly $k$ levels when $t$ is randomly cllosen from $(0, L]$ according to the $\iota 11\urcorner \mathrm{i}\mathrm{f}_{01111}$ distribution. and $\mu_{k}(c;L)$ is the k-th factorial moment of the random varible $L$ $.\backslash ^{\gamma}(\cdot:c)$ . Finally. $\rho(c;L)$ is the relative frequency of those pairs of levels below which $11\mathrm{d}\backslash I\mathrm{e}$ spacing not exceeding $c$ . We also write $\overline{\mu}_{\mathrm{A}}(c)=\lim\sup\mu_{k}(c;L)$ ; $\underline{\mu}_{k}(c)=\lim_{Larrow}\inf_{\infty}\mu k(C;L)$ ; (9) $Larrow\infty$ $\overline{\rho}(c)=\lim\sup\rho(c;L)$ ; $\underline{\rho}(c)=1\mathrm{i}\mathrm{r}\mathrm{n}\inf_{Larrow\infty}\rho(C;L)$ , (10) $Larrow\infty$ $\mathrm{d}1\overline{\perp}\mathrm{d}$ also $\pi_{k}(c)=Larrow\infty 1\mathrm{i}\ln\pi_{k}(c;L)$ ; $\mu_{k}(c)=\lim_{Larrow\infty}\mu_{k}(c;L)$ ; $\rho(c)=\lim_{Larrow\infty}\rho(c;L)$ (11) $\iota \mathrm{v}\mathrm{h}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{e}\backslash \gamma \mathrm{e}\mathrm{r}$ these lilnits exist.
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